I'd been hoping for months that someone would come along and answer this question: every time I encounter the definition of microsupport, my brain responds with a flash of anger followed by a protracted period of profound sadness. I hope that someone with a deeper understanding of this material will eventually descend from the heavens and show us a better way than the following.
Start with simplifying everything until it starts to look ridiculous. $\mathscr{F}$ is not a complex of sheaves in the derived category, but rather a single sheaf (which, if you are addicted to complexes, you can think of as being concentrated in degree zero). And rather than taking values in the category of $R$-modules for a hideous ring $R$, let's just assume we are dealing with vector spaces over a nice field ($\mathbb{Q,R,C} \ldots)$. I'll assume that you can "visualize" things like the cohomology $H^\bullet(U;\mathscr{F})$ for open sets $U \subset X$ --- this is already difficult unless you are used to thinking of sheaves in terms of their leaf spaces.
What we know for certain is that the microsupport $\text{SS}(\mathscr{F})$ lives inside the cotangent bundle $T^*X$. If you'll allow me the usual conflation of a vector space with its dual (through an inner product structure), then we can try to visualize microsupports in the tangent bundle. This contains all pairs $(x,v)$ where $x$ is a point of $X$ while $v$ lies in the tangent plane $T_xX$. If you flatten everything out by passing to a local chart around $x$, the picture is something like this:
The important point is that $v$ describes a half-space in the usual manner, consisting of all vectors in $T_xX$ whose inner product with $v$ is non-negative; and intersections of open sets $U$ around $x$ with this half-space will determine whether or not our pair $(x,v)$ lies in the microsupport $\text{SS}(\mathscr{F})$. Here is a picture:
Writing $U_v$ for the intersection of $U$ with the interior of our half-space, the cohomology group of interest is $H^\bullet_c(U_v;\mathscr{F})$, where the subscript $c$ indicates compact support. One then takes a limit over decreasing $U$. If this limiting cohomology group is zero, then the "directional derivative" of the sheaf $\mathscr{F}$ along $v$ is trivial, meaning that the sheaf propagates infinitesimally without loss of information along $v$. In this case, $(x,v)$ is not in the microsupport. But if that limiting cohomology group is not zero, then there is a phase transition in the overlaid algebraic data, in which case $(x,v)$ lies in $\text{SS}(\mathscr{F})$.
Hope this helps.